Navier–Stokes Volume V: Temporal Dynamics of High–Vorticity Interface Geometry and Empirical Subcriticality in Taylor–Green Flow

This record contains Navier–Stokes Volume V (main manuscript) and its full reproducibility supplement. Main result (Volume V): Using Taylor–Green (TG) vorticity snapshots, we measure a tube–volume (Minkowski-type) effective interface dimension d_eff(t) for high–vorticity regions. Across t ∈ [0, 20], the interface geometry remains persistently subcritical (d_eff(t) < 3) and the conclusion is robust under systematic scans of the threshold quantile q and tube-fit window ρ. Method summary: For each snapshot we define Ω(t) = {x : |ω(x,t)| ≥ Λ(t)} using a fixed quantile threshold Λ(t) = Quantile_q(|ω(·,t)|). We build a discrete interface ∂Ω(t) (morphological boundary), compute tube volumes V_ρ(t) = Vol{x : dist(x, ∂Ω(t)) ≤ ρ}, and fit log V_ρ(t) ≈ β(t) log ρ + c(t), yielding d_eff(t) = 3 − β(t). Reproducibility (Supplementary Notes, Vol. 5): The supplement provides the complete file manifest, full time-series tables, scan tables, and figures (embedded as LaTeX rows to avoid fragile CSV dependencies). It is intended as the complete computational record referenced by the main manuscript. Scope note: The TG findings are flow- and resolution-specific and do not constitute a proof of global regularity. They provide a reproducible benchmark regime consistent with dynamically maintained geometric sparseness within the Vol. I–IV geometric depletion program.